2 ORIGINS / DEVELOPMENT OF MDSMDS (aka “Smallest Space Analysis”)Has origins in Psychometrics in 1920-’60s:Scale construction and dimensionality reductionUnderwent major burst of development in 1960s due to “non-metric revolution”(Coombs) and computing developments allowing iterative estimationOriginally designed for analysis of LTM of dis/similarities data , taking a range of measures (not just PM correlations):“anything which, by an act of faith, can be considered a similarity” (Shepard)Extended rapidly to deal with wide range of other types of dataRectangular matrices ; triads, pair-comparisons, free-sorting“stacks” of matrices (3-way scaling – INDSCAL)What is MDS? Prof APM Coxon, Cardiff UU Winchester 12/09

3 Yes! And it works!!   CONSTRUCTING A MAP …Given a map, it’s easy to calculate the distances between the points …MDS operates the other way round:Given the data [ interpreted as quasi “distances” ] it attempts to find the configuration [location of points] which generated the distancesThis is “Classic MDS”: developed in 1930s – but imperfect, not robust, & works only if data are ratio.Whereas more recent MDS can work when only the ordinal information exists: “Non-metric” = ordinal MDS (Coombs / Kruskal “non-metric revolution” )What?? You can create an accurate map from only the rank –order of the distances???Yes! And it works!!  

5 WHAT IS MULTIDIMENSIONAL SCALING?A student’s definition:If you are interested in how certain objects relate to each other … and if you would like to present these relationships in the form of a map then MDS is the technique you need” (Mr Gawels, KUB) A good start!MDS provides …a useful and easily-assimilable graphic visualisation of all sorts of dataTukey: “A picture is worth a thousand words”In a user-chosen (small) # of dimensionsproviding a graphical representation of the structure underlying a complex data setAnd measure how well / badly the solution distances match the data dissimilarities (Stress)What is MDS? Prof APM Coxon, Cardiff UniUni Winchester 12/09

6 MDS is a family of models differentiated by …(DATA) the empirical inter-relationships between a set of “objects”/variables which are given in a set of dis/similarity dataBasically, type of input data, defined by their “Way” and “Mode” [e.g. 2W1M]. (Cf observations vs data)(FUNCTION) data are then optimally re- scaled (according to permissible trans- formations for the data) in terms of …Choice of level of measurement [e.g. ordinal ](MODEL) the assumptions of the model chosen to represent the dataUsually (Euclidean) Distance modelWhat is MDS? Prof APM Coxon, Cardiff UU Winchester, 12/2009

17 SOME POSSIBLE WEAKNESSES in MDS There ARE any??!Relative ignorance of the sampling/inferential properties of stressBut, simulation (Spence), MLE estimationProne-ness to local minima solutionsbut less so, and multiple starts & interactive programs like PERMAP allow thousands of runs to checkA few forms of data/models are prone to degeneraciesespecially MD Unfolding, but see new PREFSCAL in SPSS14)difficulty in representing the asymmetry of causal modelsthough external analysis is very akin to dependent-independent modelling,there are convergences with GLM in hybrid models such as CLASCAL (INDSCAL with parameterization of latent classes)What is MDS? Prof APM Coxon, U EdinburghResearch Methods Festival 2006